Optimal. Leaf size=157 \[ -\frac {2 x^3}{3 b^3 \pi ^2}+\frac {x^3 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \text {FresnelC}(b x)}{b^4 \pi ^2}+\frac {43 S\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}-\frac {8 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^4 \text {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {11 x \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6590, 6598,
6588, 3432, 3473, 30, 3467, 3466} \begin {gather*} \frac {43 S\left (\sqrt {2} b x\right )}{8 \sqrt {2} \pi ^3 b^6}-\frac {2 x^3}{3 \pi ^2 b^3}+\frac {x^4 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {8 \text {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^3 b^6}-\frac {11 x \sin \left (\pi b^2 x^2\right )}{8 \pi ^3 b^5}+\frac {4 x^2 \text {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi ^2 b^4}+\frac {x^3 \cos \left (\pi b^2 x^2\right )}{4 \pi ^2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3432
Rule 3466
Rule 3467
Rule 3473
Rule 6588
Rule 6590
Rule 6598
Rubi steps
\begin {align*} \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx &=\frac {x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {4 \int x^3 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^2 \pi }-\frac {\int x^4 \sin \left (b^2 \pi x^2\right ) \, dx}{2 b \pi }\\ &=\frac {x^3 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac {x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {8 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x) \, dx}{b^4 \pi ^2}-\frac {3 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{4 b^3 \pi ^2}-\frac {4 \int x^2 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}\\ &=\frac {x^3 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}-\frac {8 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {3 x \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {3 \int \sin \left (b^2 \pi x^2\right ) \, dx}{8 b^5 \pi ^3}+\frac {4 \int \sin \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}-\frac {2 \int x^2 \, dx}{b^3 \pi ^2}-\frac {2 \int x^2 \cos \left (b^2 \pi x^2\right ) \, dx}{b^3 \pi ^2}\\ &=-\frac {2 x^3}{3 b^3 \pi ^2}+\frac {x^3 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac {3 S\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}+\frac {2 \sqrt {2} S\left (\sqrt {2} b x\right )}{b^6 \pi ^3}-\frac {8 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {11 x \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}+\frac {\int \sin \left (b^2 \pi x^2\right ) \, dx}{b^5 \pi ^3}\\ &=-\frac {2 x^3}{3 b^3 \pi ^2}+\frac {x^3 \cos \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {4 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)}{b^4 \pi ^2}+\frac {11 S\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^6 \pi ^3}+\frac {2 \sqrt {2} S\left (\sqrt {2} b x\right )}{b^6 \pi ^3}-\frac {8 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^6 \pi ^3}+\frac {x^4 C(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^2 \pi }-\frac {11 x \sin \left (b^2 \pi x^2\right )}{8 b^5 \pi ^3}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 120, normalized size = 0.76 \begin {gather*} \frac {-32 b^3 \pi x^3+12 b^3 \pi x^3 \cos \left (b^2 \pi x^2\right )+129 \sqrt {2} S\left (\sqrt {2} b x\right )+48 \text {FresnelC}(b x) \left (4 b^2 \pi x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )+\left (-8+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )-66 b x \sin \left (b^2 \pi x^2\right )}{48 b^6 \pi ^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.95, size = 202, normalized size = 1.29
method | result | size |
default | \(\frac {\frac {\FresnelC \left (b x \right ) \left (\frac {b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }-\frac {4 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{\pi }\right )}{b^{5}}-\frac {\frac {2 b^{3} x^{3}}{3 \pi ^{2}}+\frac {\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{\pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{2 \pi }}{\pi ^{2}}+\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \mathrm {S}\left (b x \sqrt {2}\right )}{2 \pi ^{3}}}{b^{5}}}{b}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 132, normalized size = 0.84 \begin {gather*} \frac {24 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 44 \, \pi b^{4} x^{3} + 192 \, \pi b^{3} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) + 129 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) - 12 \, {\left (11 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 4 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{48 \, \pi ^{3} b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^5\,\mathrm {FresnelC}\left (b\,x\right )\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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